| L(s) = 1 | + (−1.52 − 1.29i)2-s + (3.39 − 2.78i)3-s + (0.626 + 3.95i)4-s + (−2.17 − 0.661i)5-s + (−8.77 − 0.171i)6-s + (−1.02 − 5.14i)7-s + (4.17 − 6.82i)8-s + (2.00 − 10.0i)9-s + (2.45 + 3.83i)10-s + (−1.58 − 16.1i)11-s + (13.1 + 11.6i)12-s + (3.26 + 10.7i)13-s + (−5.12 + 9.15i)14-s + (−9.23 + 3.82i)15-s + (−15.2 + 4.94i)16-s + (0.474 − 1.14i)17-s + ⋯ |
| L(s) = 1 | + (−0.760 − 0.649i)2-s + (1.13 − 0.928i)3-s + (0.156 + 0.987i)4-s + (−0.435 − 0.132i)5-s + (−1.46 − 0.0286i)6-s + (−0.146 − 0.735i)7-s + (0.522 − 0.852i)8-s + (0.222 − 1.11i)9-s + (0.245 + 0.383i)10-s + (−0.144 − 1.46i)11-s + (1.09 + 0.971i)12-s + (0.251 + 0.828i)13-s + (−0.366 + 0.654i)14-s + (−0.615 + 0.254i)15-s + (−0.951 + 0.309i)16-s + (0.0278 − 0.0673i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.626 + 0.779i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.626 + 0.779i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.525908 - 1.09754i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.525908 - 1.09754i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.52 + 1.29i)T \) |
| good | 3 | \( 1 + (-3.39 + 2.78i)T + (1.75 - 8.82i)T^{2} \) |
| 5 | \( 1 + (2.17 + 0.661i)T + (20.7 + 13.8i)T^{2} \) |
| 7 | \( 1 + (1.02 + 5.14i)T + (-45.2 + 18.7i)T^{2} \) |
| 11 | \( 1 + (1.58 + 16.1i)T + (-118. + 23.6i)T^{2} \) |
| 13 | \( 1 + (-3.26 - 10.7i)T + (-140. + 93.8i)T^{2} \) |
| 17 | \( 1 + (-0.474 + 1.14i)T + (-204. - 204. i)T^{2} \) |
| 19 | \( 1 + (-2.81 + 5.25i)T + (-200. - 300. i)T^{2} \) |
| 23 | \( 1 + (-8.00 + 5.34i)T + (202. - 488. i)T^{2} \) |
| 29 | \( 1 + (2.14 - 21.8i)T + (-824. - 164. i)T^{2} \) |
| 31 | \( 1 + (-11.2 - 11.2i)T + 961iT^{2} \) |
| 37 | \( 1 + (-18.0 - 33.7i)T + (-760. + 1.13e3i)T^{2} \) |
| 41 | \( 1 + (-55.3 + 36.9i)T + (643. - 1.55e3i)T^{2} \) |
| 43 | \( 1 + (-48.4 - 39.7i)T + (360. + 1.81e3i)T^{2} \) |
| 47 | \( 1 + (14.5 - 35.1i)T + (-1.56e3 - 1.56e3i)T^{2} \) |
| 53 | \( 1 + (9.46 + 96.1i)T + (-2.75e3 + 548. i)T^{2} \) |
| 59 | \( 1 + (-83.0 - 25.1i)T + (2.89e3 + 1.93e3i)T^{2} \) |
| 61 | \( 1 + (86.0 - 70.6i)T + (725. - 3.64e3i)T^{2} \) |
| 67 | \( 1 + (6.03 + 7.34i)T + (-875. + 4.40e3i)T^{2} \) |
| 71 | \( 1 + (-120. + 23.8i)T + (4.65e3 - 1.92e3i)T^{2} \) |
| 73 | \( 1 + (-4.09 - 0.815i)T + (4.92e3 + 2.03e3i)T^{2} \) |
| 79 | \( 1 + (-6.00 - 14.4i)T + (-4.41e3 + 4.41e3i)T^{2} \) |
| 83 | \( 1 + (135. + 72.5i)T + (3.82e3 + 5.72e3i)T^{2} \) |
| 89 | \( 1 + (11.6 - 17.3i)T + (-3.03e3 - 7.31e3i)T^{2} \) |
| 97 | \( 1 + (56.8 - 56.8i)T - 9.40e3iT^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.82765124511435909102992592620, −11.64500373822448960797604406172, −10.73795963869305996712162517806, −9.287365639566989650433989210734, −8.456908708659228830652360171203, −7.69587822886284790477603117820, −6.66500272232081244797368707848, −3.92495906904310639672609985939, −2.76459893928474194418667592181, −1.01502239269709497401073748860,
2.48365622259383705226029961285, 4.23506559868710812377653248910, 5.69564138691333162174057606896, 7.42958541100607536322272897704, 8.199686402337240864565887361571, 9.351106460311686532325556264304, 9.831983371575754843285910423218, 11.00044910900630101282616675287, 12.48484081678275689403094011659, 13.91384949223861422046853587722
